Finding the maximum of a index that involves a distance matrix

Dear all,

We are urban/regional economists in Brazil and we are working on a index of urban monocentricity. We want to improve the "spatial separation index" (SP) (http://www.princeton.edu/~reddings/papers/dgII_loc.pdf see 3.4) in order to reach our aims. However, we are stuck in the following issue:
As part of our new index, we need to calculate the maximum value that the SP index can reach at each map.
The SP index goes like this in a city with n spatial units:

SP= S' . D . S
S: vector (n X 1), of the share of each spatial unit in the total population of the city.
D: Euclidian distance between each area i and j. (The main diagonal is 0).

We have ran simulations and the maximum of SP seems to be when each extreme spatial unit has a fourth of the population ( i.e. 1/4 of the population in the spatial unit on the north, 1/4 on the south, 1/4 on the east, 1/4 on the west).
However we do not have a general (analytical ) solution to the problem. (In fact, we do not know even if our "solution" is true!) . In other words, what is the maximum of SP?
Obviously we will acknowledge your help in our paper.
Yours thankfully,
L Monasterio.
PS. Sorry if this is not the right sub-forum. I did my best to find the right one.